快速选择算法因元素重复而失败

quickselect algorithm fails with duplicated elements

本文关键字：失败 元素 算法 快速选择      更新时间：2023-10-16

我已经实现了快速选择算法。

我有一个问题，当我在数组中使用重复项时，我的算法最终会陷入无休止的循环。。。

你能帮我把它修好吗？

期望的复杂度是O(n)与最坏情况O(n^2)？

#include <iostream> 
#include <vector> 
#include <algorithm> 
#include <ctime> 
using namespace std; 
int rand_partition(vector<int> &a, int left, int right) { 
int pivotIndex = left + (rand() % (right - left)); 
//int m = left + (right - left) / 2; //... to test the algo...no rand at this point 
int pivot = a[pivotIndex]; 
int i = left; 
int j = right; 
do { 
while (a[i] < pivot) i++; // find left element > pivot 
while (a[j] > pivot) j--; // find right element < pivot 
// if i and j not already overlapped, we can swap 
if (i < j) { 
swap(a[i], a[j]); 
} 
} while (i < j); 
return i; 
} 
// Returns the n-th smallest element of list within left..right inclusive (i.e. n is zero-based). 
int quick_select(vector<int> &a, int left, int right, int n) { 
if (left == right) {        // If the list contains only one element 
return a[left];  // Return that element 
} 
int pivotIndex = rand_partition(a, left, right); 
// The pivot is in its final sorted position 
if (n == pivotIndex) { 
return a[n]; 
} 
else if (n < pivotIndex) { 
return quick_select(a, left, pivotIndex - 1, n); 
} 
else { 
return quick_select(a, pivotIndex + 1, right, n); 
} 
} 
int main() { 
vector<int> vec= {1, 0, 3, 5, 0, 8, 6, 0, 9, 0}; 
cout << quick_select(vec, 0, vec.size() - 1, 5) << endl; 
return 0; 
}

您的代码中有几个问题。

  • 首先，在函数quick_select()中，直接比较pivotIndex和n。由于left并不总是0，您应该将n与左侧部分的长度进行比较，后者等于pivotIndex - left + 1
  • 当n > length时，你只需要递归地调用quick_select(a, pivotIndex + 1, right, n)，此时，它意味着整个向量的第N个元素位于它的右部分，它是向量右部分的第(N-(pivotIndex-left+1))个元素代码应该是quick_select(a, pivotIndex + 1, right, n - (pivotIndex - left + 1) )(n是基于ONE的)
  • 您似乎使用了霍尔的分区算法，并且实现错误。即使它有效，当HOARE-PARTITION终止时，它也会返回一个值j，即A[p...j] ≤ A[j+1...r]，但我们希望quick_select()中有A[p...j-1] ≤ A[j] ≤ A[j+1...r]。所以我使用了基于我在另一篇文章中写的Lomuto分区算法的rand_partition()

这里是固定的quick_select()，它返回向量的第N个(注意，N是基于一的)最小元素：

int quick_select(vector<int> &a, int left, int right, int n)
{
if ( left == right ) 
return a[left];
int pivotIndex = partition(a, left, right);
int length = pivotIndex - left + 1;
if ( length == n) 
return a[pivotIndex];
else if ( n < length ) 
return quick_select(a, left, pivotIndex - 1, n);
else 
return quick_select(a, pivotIndex + 1, right, n - length);
}

这就是rand_partition():

int rand_partition(vector<int> &arr, int start, int end)
{
int pivot_index = start + rand() % (end - start + 1);
int pivot = arr[pivot_index];
swap(arr[pivot_index], arr[end]); // swap random pivot to end.
pivot_index = end;
int i = start -1;
for(int j = start; j <= end - 1; j++)
{
if(arr[j] <= pivot)
{
i++;
swap(arr[i], arr[j]);
}
}
swap(arr[i + 1], arr[pivot_index]); // swap back the pivot
return i + 1;
}

首先调用srand()来初始化随机数生成器，以便在调用rand()时可以获得类似随机数的数字
测试上述功能的驱动程序：

int main()
{
int A1[] = {1, 0, 3, 5, 0, 8, 6, 0, 9, 0};
vector<int> a(A1, A1 + 10);
cout << "6st order element " << quick_select(a, 0, 9, 6) << endl;
vector<int> b(A1, A1 + 10); // note that the vector is modified by quick_select()
cout << "7nd order element " << quick_select(b, 0, 9, 7) << endl;
vector<int> c(A1, A1 + 10);
cout << "8rd order element " << quick_select(c, 0, 9, 8) << endl;
vector<int> d(A1, A1 + 10);
cout << "9th order element " << quick_select(d, 0, 9, 9) << endl;
vector<int> e(A1, A1 + 10);
cout << "10th order element " << quick_select(e, 0, 9, 10) << endl;
}
相关文章：